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Stress Intensity Factor

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Introduction

The stress intensity factor was developed in 1957 by George R Irwin, the man usually considered to be the father of fracture mechanics [1]. The stress intensity factor is abbreviated SIF and represented by the variable, \(K\). It is one of the most fundamental and useful parameters in all of fracture mechanics. The stress intensity factor describes the stress state at a crack tip, is related to the rate of crack growth, and is used to establish failure criteria due to fracture.

Irwin arrived at the definition of \(K\) as a near-crack-tip approximation to Westergaard's complete solution for the stress field surrounding a crack [2]. Recall that Westergaard used complex numbers and Airy stress functions to do so. We will first review Westergaard's solution, and then see how Irwin used it to develop the stress intensity factor.

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Westergaard's Solution

Westergaard found an Airy stress function of complex numbers that is the solution for the stress field in an infinite plate containing a crack.

The complete set of equations for the stress field is

\[ \begin{eqnarray} \sigma_{xx} & = & \text{Re}\,Z - y \, \text{Im} \, Z' \\ \\ \sigma_{yy} & = & \text{Re}\,Z + y \, \text{Im} \, Z' \\ \\ \tau_{xy} & = & - y \, \text{Re} \, Z' \end{eqnarray} \]
where \(Z\) and its derivative, \(Z'\), are

\[ Z(z) = {\sigma_\infty \over \sqrt{1 - \left( a \over z \right)^2}} \quad \qquad \text{and} \qquad \quad Z'(z) = {- \sigma_\infty \, a^2 \over z^3 \left[ 1 - \left( a \over z \right)^2 \right]^{3/2} } \]
and \(a\) is crack length, and \(z\) equals \(x + i y\).

When \(y = 0\), the equation for \(\sigma_{yy}\) reduces to

\[ \sigma_{yy} \; = \; { \sigma_\infty \over \sqrt{1 - \left( a \over x \right)^2} } \]
A plot is shown below. It shows the stress value quickly dropping from infinity at the crack tip, \(x = a\), to the far-field value of \(\sigma_\infty\). Keep in mind that this is along the crack plane, \(y=0\).



The equation and its graph are the key results of Westergaard's solution that are shared by most authors, and with good reason. Computing the stress at any other position near the crack requires a Taylor series expansion of the function in order to partition it into its real and imaginary parts... a great deal of work. This imposes a significant hurdle to the general understanding of the stress state surrounding a crack.

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Irwin's Near Crack Tip Solution

Two decades later, Irwin showed that Westergaard's result could be greatly simplified in the area immediately surrounding the crack tip. He did so by expressing \(z\) as the sum of two terms.

\[ z = a + r e^{i \theta} \]
The key feature of this expression is that \(r = 0\) at the crack tip (\(x=a\)). This is reflected by the red components in the figure. And note that \(r \lt \lt a\) in the region very near the crack tip. This inequality was used by Irwin to find simple expressions (well, compared to the complex functions) for the stress components near the crack tip in terms of polar coordinates, \(r\) and \(\theta\). And we will see how, along the way, the definition of the stress intensity factor naturally falls out.

Evaluating the Z(z) Term

Substituting \(z = a + r e^{i \theta}\) into \(Z(z)\) gives

\[ Z(z) = {\sigma_\infty \over \sqrt{1 - \left( a \over a + r e^{i \theta} \right)^2}} \]
and applying some algebra, specifically obtaining a common denominator, gives

\[ Z(z) = {\sigma_\infty \over \sqrt{ 2 a r e^{i \theta} + r^2 e^{i 2 \theta} \over a^2 + 2 a r e^{i \theta} + r^2 e^{i 2 \theta} }} \]
It's at this point that Irwin imposed some physical insight on the problem in order to simplify the equation. He recognized that the area immediately near the crack tip corresponds to very small values of \(r\), so small in fact that \(a \gt \gt r\). (This is exactly why he expressed \(z\) as \(a + r e^{i \theta}\), after all.) Therefore, \(a^2 \gt \gt a \, r \gt \gt r^2\), and all terms following the first in both the numerator and denominator can be neglected, leaving

\[ Z(z) = {\sigma_\infty \over \sqrt{ 2 \, a \, r \, e^{i \theta} \over a^2 } } \]
which can be further simplified to

\[ Z(z) = \sigma_\infty \sqrt{ a \over 2 \, r } \; e^{-i {\theta \over 2} } \]
Euler's famous identity, \(e^{i \phi} = \cos \phi + i \, \sin \phi\), is next used to get

\[ Z(z) = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - i \, \sin {\theta \over 2} \right) \]
And since Westergaard's solution only needs the real part of \(Z(z)\), then

\[ \text{Re} \, Z = \sigma_\infty \sqrt{ a \over 2 \, r } \; \cos {\theta \over 2} \]

Evaluating the Z'(z) Term

The process is repeated for the derivative, \(Z'\). The \(a + r e^{i \theta}\) expression is first substituted for \(z\). Then negligible terms are neglected because \(a \gt \gt r\). Finally, Euler's identity is used to separate the result into real and imaginary parts.

\[ \begin{eqnarray} Z'(z) & = & {- \sigma_\infty \, a^2 \over z^3 \left[ 1 - \left( a \over z \right)^2 \right]^{3/2} } \\ \\ & = & {- \sigma_\infty \, a^2 \over (a + r e^{i \theta})^3 \left[ 1 - \left( a \over (a + r e^{i \theta}) \right)^2 \right]^{3/2} } \\ \\ & = & {- \sigma_\infty \, a^2 \over (a + r e^{i \theta})^3 \left[ 2 a r e^{i \theta} + r^2 e^{i 2 \theta} \over a^2 + 2 a r e^{i \theta} + r^2 e^{i 2 \theta} \right]^{3/2} } \\ \end{eqnarray} \]
And neglect negligible terms again because \(a \gt \gt r\).

\[ Z'(z) = {- \sigma_\infty \, a^2 \over a^3 \left[ 2 a r e^{i \theta} \over a^2 \right]^{3/2} } \]
This simplifies to

\[ Z'(z) = {- \sigma_\infty \over 2 \, r} \sqrt{ a \over 2 \, r} \; e^{-i {3 \theta \over 2}} \]
Using the Euler identity gives

\[ Z'(z) = {- \sigma_\infty \over 2 \, r} \sqrt{ a \over 2 \, r} \; \left( \cos {3 \theta \over 2} - i \sin {3 \theta \over 2} \right) \]

Expressions for Stress Components

Now that the real and imaginary parts of \(Z\) and \(Z'\) have been determined, it is possible to write out expressions for the stress components. Beginning with \(\sigma_{xx}\), recall that

\[ \sigma_{xx} = \text{Re}\,Z - y \, \text{Im} \, Z' \]
Inserting the appropriate real and imaginary components gives Irwin's near crack tip approximation

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \cos {\theta \over 2} - y \; {\sigma_\infty \over 2 \, r} \; \sqrt{ a \over 2 \, r} \; \sin {3 \theta \over 2} \]
which simplifies to

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - {y \over 2 \, r} \; \sin {3 \theta \over 2} \right) \]
And \(y\) can be replaced by \(r \sin \theta\). This gives

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - {1 \over 2} \; \sin \theta \; \sin {3 \theta \over 2} \right) \]
The following trig identity is used next: \({1 \over 2} \sin \theta = \sin {\theta \over 2} \cos {\theta \over 2}\)

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - \sin {\theta \over 2} \cos {\theta \over 2} \sin {3 \theta \over 2} \right) \]
which permits a \(\cos {\theta \over 2}\) to be factored out of both terms.

\[ \sigma_{xx} = {\sigma_\infty \sqrt{a} \over \sqrt{ 2 r }} \cos {\theta \over 2} \left( 1 - \sin {\theta \over 2} \sin {3 \theta \over 2} \right) \]
Irwin's development stopped here, and this is indeed the natural place to stop. Nevertheless, in the years following Irwin's publication, it became popular to include \(\sqrt{\pi/\pi}\) in the expressions as follows.

\[ \sigma_{xx} = {\sigma_\infty \sqrt{\pi a} \over \sqrt{ 2 \pi r }} \cos {\theta \over 2} \left( 1 - \sin {\theta \over 2} \sin {3 \theta \over 2} \right) \]
Repeating the process for \(\sigma_{yy}\) and \(\tau_{xy}\) gives

\[ \sigma_{yy} = {\sigma_\infty \sqrt{\pi a} \over \sqrt{ 2 \pi r }} \cos {\theta \over 2} \left( 1 + \sin {\theta \over 2} \sin {3 \theta \over 2} \right) \]
and

\[ \tau_{xy} = {\sigma_\infty \sqrt{\pi a} \over \sqrt{ 2 \pi r }} \cos {\theta \over 2} \sin {\theta \over 2} \cos {3 \theta \over 2} \]

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Comparison to Exact Solution

Recall that the expressions for the stress components in terms of \(r\) and \(\theta\) involved a critical simplifying step in their development that makes them accurate only near the crack tip. The natural question to ask is how close must one be to the tip to rely on the equations. The usual method of answering this is to look at \(\sigma_{yy}\) along \(y = 0\). Recall that the exact expression is

\[ \sigma_{yy} = { \sigma_\infty \over \sqrt{1 - \left( a \over x \right)^2} } \]
and Irwin's approximation (with \(\theta = 0\)) is

\[ \sigma_{yy} = { \sigma_\infty \sqrt{ \pi a} \over \sqrt{2 \pi r} } \]
The two equations are shown in the graph below. It is clear that both are very close at the crack tip and diverge as the distance from the tip increases. The region of close agreement is approximately \(r \le a/10\). Beyond this, the approximate expression continues to decrease toward zero because \(1/\sqrt{r}\) always decreases as \(r\) increases. In contrast, the exact solution levels out at \(\sigma_\infty\).



One might ask, why even bother with Irwin's approximate solution when Westergaard's exact solution is available? There are several reasons. First, the approximate solution is indeed accurate at the crack tip, and this is all that really matters because only the conditions at the crack tip dictate (i) how fast the crack grows, (ii) in which direction it grows, and (iii) whether or not it fails catastrophically. Second, the approximate solution clearly reveals the dependence of the stress components on \(r\) and \(\theta\), something that is present, but masked in complexity in the exact solution. Finally, the approximate solution leads to the definition of the stress intensity factor, K, one of the most important parameters in all of fracture mechanics. It is explained in the next section.

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Stress Intensity Factor, K

Note that the numerators of all three stress equations contain the same expression, \(\sigma_\infty \sqrt{\pi a}\). This combination of \(\sigma_\infty\) and \(a\) completely describes the severity of the stress state at the crack tip. Irwin recognized this and first used the term stress intensity factor to describe the expression. (Although he didn't include \(\pi\) in the beginning. It was added later.)

\[ K = \sigma_\infty \sqrt{\pi a} \]
Note also that all three expressions have a \(\sqrt{r}\) in the denominator, which dictates the dependence of stress on distance from the crack tip and reflects the singularity at \(r=0\). And finally, all three expressions contain functions of \(\theta\) alone, not intermingled with \(r\), \(\sigma_\infty\), or \(a\). Therefore, all three expressions can be written compactly as

\[ \sigma_{xx} = {K \over \sqrt{ 2 \pi r }} f(\theta) \qquad \qquad \sigma_{yy} = {K \over \sqrt{ 2 \pi r }} g(\theta) \qquad \qquad \tau_{xy} = {K \over \sqrt{ 2 \pi r }} h(\theta) \]

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Relationship to Energy Release Rate

This topic was covered on the page on Griffith's energy release rate, but will be reviewed again here [3]. Recall that his critical value, \(\mathcal{G}_c\), was related to the failure stress, \(\sigma_f\), by

\[ \mathcal{G}_c \; = \; { \sigma_f^2 \, \pi \, a \over E} \]
The numerator is the square of the critical Stress Intensity Factor, \(K_c\), defined as

\[ K_c = \sigma_f \sqrt{\pi \, a} \]
and this means that \(\mathcal{G}_c\) and \(K_c\) are related according to

\[ \mathcal{G}_c = { \, K_c^2 \over E \,} \]
More generally, noncritical values of \(\mathcal{G}\) and \(K\) are related in the same way

\[ \mathcal{G} = { \, K^2 \over E \,} \]
when \(\sigma\) is less than \(\sigma_f\).

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References

1. Irwin, G.R., "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate," Journal of Applied Mechanics, Vol. 24, pp. 361-364, 1957.
2. Westergaard, H.M., "Bearing Pressures and Cracks," Journal of Applied Mechanics, Vol. 6, pp. A49-53, 1939.
3. Griffith, A.A., "The Phenomena of Rupture and Flow in Solids," Philosophical Transactions, Series A, Vol. 221, pp. 163-198, 1920.